Yes you can send a digital signal by any modulation scheme you like.
Analog modems (at least the old 9600baud ones) are purely AM and regular FM radio includes a digital data channel for traffic reports

Since you're transmitting binary, there are two levels in either case. So, it's Amplitude Shift Modulation (shift between two levels) or Frequency Shift Modulation (shift between two frequencies).

As for phase shift modulation, I've only seen it used for digital and it get pretty complicated.

Of the various techniques, FSK is my favorite. It's simple and handles noise well. In any of the cases, there are issues restoring the data on the other end. So, you're better off using someone else's modules.

You can find modules at DigiKey, or check out microchip.com for FSK chips.

There is also AFSK which is technically what the original telephone modems were. Two audio frequencies are picked and they are shifted back and forth similar to FSK. This audio signal is then sent down a phone line, or sent into the modulator of your choice. It could be AM or FM.

You should realise that any transmitted signal is, in fact, an analog signal. Some property of a real e.m. wave is being varied - rather than 'switched' ,which takes up far too much bandwidth.. The square 'box-car' waves which you see in all the books seldom exist in practice. If you can transmit a nice looking waveform like that in your available channel space then you are seriously wasting good spectrum space.
'Real' digitally modulated signals are very 'rounded off' and the symbols all interfere with each other in an alarming way.
You can just A.M. a transmitter with a low pass filtered binary (or N'ary) signal and it can be received and decoded very easily. The only snag is that it is rather wasteful of power and spectrum space.
Likewise, an FM transmitter can be fed with a similar signal and moduated with a low modulation index.
I was interested to read the comment above that phase modulation is difficult. If you take a frequency modulator and feed it with a differentiated version of the data waveform then you have phase modulation. Indeed, broadcast 'FM' is effectively PM, actually because of the pre-emphasis which is performed on the signal prior to modulation.
Of course, to get the best out of a channel, you wouldn't do your PM (which is what your PSK signal really is) in that way but it can still be done if you have plenty of channel space and carrier strength.